Without more details, it's hard to judge. My hand was mauled by a feral
cat at a colony I cared for. It was my fault, my mistake. I agitated and
apparently frightened the cat unintentionally while he was eating.

There was a lot of blood when he was done with me. At one point, I
literally felt his teeth sink into the knuckle bones of my left thumb. I
had to throw him to get him let go (he was alright, but not me). When I
finally recovered sufficient to get home, I didn't do anything but the
sort of stuff you did; wash the wounds, attempt to sterilize, bandage
and try to ignore how much it hurt. I was up-to-date on my rabies
vaccinations, so I thought I was safe.

Later I found out that there can be very severe consequences from animal
bites, not the least of which is fatal septic shock syndrome that can
occur quite some time after the attack. The infections that can occur
are terrifying. I won't presume to tell you what to do, but if I ever
get hurt like that again, I will go straight to the emergency room.

Another issue to consider is that if the animal is available,  the
attack will be reported and, depending on certain criteria, authorities
may kill the animal to determine if it is rabid. This was another reason
why I didn't go to the doctor. I knew that the cat was not rabid (I had
earlier trapped, tested, vaccinated and neutered him). Stupid though it
may sound to some people, I didn't want to risk that he would be killed
for something that was my own damned fault.

I'm a lot more wary when dealing with unknown animals now.